In the series De Gruyter Textbook
This book covers results in the qualitative theory of ODEs and PDEs and presents several techniques for their solution. It covers both linear and nonlinear differential equations of first and higher order, systems of differential equations, Sturm-Liouville problems, Bessel functions, and approximate and numerical methods. The 2nd edition includes new applied examples, and exercises and a revised chapter on Laplace Transforms and their inverses.
Covers solving linear and nonlinear equations and systems
Discusses theoretical aspects and qualitative analysis
New material includes a comprehensive table of the inverses of the Laplace transform
Additional applied examples and exercises
Antonio Ambrosetti, SISSA, Italy; Shair Ahmad, University of Texas at San Antonio, USA.
Analysis
Differential Equations and Dynamical Systems
Mathematics
Large scale geometry is the study of geometric objects viewed from a great distance. The idea of large scale geometry can be traced back to Mostowfs work on rigidity and the work of ?varc, Milnor and Wolf on growth of groups and is greatly influenced by Gromovfs work on geometric group theory. In the last decades, large scale geometry has found important applications in group theory, topology, geometry, higher index theory, computer science, and large data analysis.
This book provides a friendly approach to the basic theory of this exciting and fast growing subject and offers a glimpse of its applications to topology, geometry, and higher index theory. The authors have made a conscientious effort to make the book accessible to advanced undergraduate students, graduate students, and non-experts.
The present second edition has been updated to cover recent developments involving constructions of groups and metric spaces with exotic properties as well as results charting new directions in index theory, and it also includes minor improvements in the presentation and an updated bibliography.
Download pp. xi?xiii
Notation and conventions
pp. xv?xvi
1 Metric spaces and large scale geometry
pp. 1?26
2 Asymptotic dimension and decomposition complexity
pp. 27?49
3 Amenability
pp. 51?64
4 Property A
pp. 65?85
5 Coarse embeddings
pp. 87?122
6 Group actions on Banach spaces
pp. 123?146
7 Coarse homology
pp. 147?169
8 Survey of applications
pp. 171?177
References
pp. 179?194
Index
pp. 195?197
The theory of buildings lies at the interplay between geometry and group theory, and is one of the main tools for studying the structure of many groups.
Actually, buildings were introduced by Jacques Tits in the 1950s to better understand and study a semi-simple algebraic group over a field. For a general field, its associated building is a spherical building, called its Tits building. It is a simplicial complex and, in this book, one considers a geometric realization called vectorial building. When the field is real valued, Francois Bruhat and Jacques Tits constructed another building taking into account the topology of the field. This Bruhat?Tits building is a polysimplicial complex and its usual geometric realization is an affine building.
These vectorial or affine buildings are the main examples of Euclidean buildings. The present book develops the general abstract theory of these Euclidean buildings (the buildings with Euclidean affine spaces as apartments). It is largely self-contained and emphasizes the metric aspects of these objects, as CAT(0) spaces very similar to Riemannian symmetric spaces of non-compact type. The book studies their compactifications, their links with groups, many classical examples, and some applications (for example, to Hecke algebras).
Download pp. ix?x
Introduction
Download pp. 1?8
1 Euclidean apartments
pp. 9?76
2 Euclidean buildings
pp. 77?138
3 Properties at infinity
pp. 139?208
4 Groups of automorphisms
pp. 209?254
5 Buildings associated to groups
pp. 255?325
6 Examples
pp. 327?451
7 Applications
pp. 453?567
A Notations
pp. 569?571
References
pp. 573?583
Notation index
pp. 585?588
Subject index
pp. 589?597
This book treats the classical problem, posed by Kazdan and Warner, of prescribing a given function on a closed manifold as the scalar curvature of a metric within a conformal class. Since both critical equations and obstructions to the existence of solutions appear, the problem is particularly challenging.
Our focus is to present a general approach for understanding the matter, particularly the issue of loss of compactness. The task of establishing existence of solutions is attacked combining several tools: the variational structure of the problem, Liouville-type theorems, blow-up analysis, elliptic regularity theory, and topological arguments.
Treating different aspects of the subject and containing several references to up-to-date research directions and perspectives, the book will be useful to graduate students and researchers interested in geometric analysis and partial differential equations.
Download pp. ix?x
1 Introduction
Download pp. 1?10
2 Preliminaries
pp. 11?21
3 The Yamabe quotient and resolution of (KW-1)
pp. 23?39
4 Variational structure of (KW-2)
pp. 41?50
5 Classification of entire solutions in the Euclidean space
pp. 51?59
6 Blow-up analysis
pp. 61?76
7 Finite-dimensional reduction of
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,
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)
(E
K,Ñ
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)
pp. 77?89
8 General existence results for
(
E
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)
(E
K
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)
pp. 91?105
9 Forbidden curvatures
pp. 107?118
10 Related directions and perspectives
pp. 119?129
A Some useful results
pp. 131?138
References
pp. 139?151
ISBN: 978-1-119-89181-9 August 2023 240 Pages
M-statistics introduces a new approach to statistical inference, redesigning the fundamentals of statistics, and improving on the classical methods we already use. This book targets exact optimal statistical inference for a small sample under one methodological umbrella. Two competing approaches are offered: maximum concentration (MC) and mode (MO) statistics combined under one methodological umbrella, which is why the symbolic equation M=MC+MO. M-statistics defines an estimator as the limit point of the MC or MO exact optimal confidence interval when the confidence level approaches zero, the MC and MO estimator, respectively. Neither mean nor variance plays a role in M-statistics theory.
Novel statistical methodologies in the form of double-sided unbiased and short confidence intervals and tests apply to major statistical parameters:
Exact statistical inference for small sample sizes is illustrated with effect size and coefficient of variation, the rate parameter of the Pareto distribution, two-sample statistical inference for normal variance, and the rate of exponential distributions.
M-statistics is illustrated with discrete, binomial, and Poisson distributions. Novel estimators eliminate paradoxes with the classic unbiased estimators when the outcome is zero.
Exact optimal statistical inference applies to correlation analysis including Pearson correlation, squared correlation coefficient, and coefficient of determination. New MC and MO estimators along with optimal statistical tests, accompanied by respective power functions, are developed.
M-statistics is extended to the multidimensional parameter and illustrated with the simultaneous statistical inference for the mean and standard deviation, shape parameters of the beta distribution, the two-sample binomial distribution, and finally, nonlinear regression.
Our new developments are accompanied by respective algorithms and R codes, available at GitHub, and as such readily available for applications.
M-statistics is suitable for professionals and students alike. It is highly useful for theoretical statisticians and teachers, researchers, and data science analysts as an alternative to classical and approximate statistical inference.
1st Edition - January 1, 2024
Paperback ISBN: 9780443236013
Fractional Difference, Differential Equations, and Inclusions: Analysis and Stability is devoted to the existence and stability (Ulam-Hyers-Rassias stability and asymptotic stability) of solutions for several classes of functional fractional difference equations and inclusions. Covered equations include delay effects of finite, infinite, or state-dependent nature, and tools used to establish the existence results for the proposed problems include fixed point theorems, densifiability techniques, monotone iterative technique, notions of Ulam stability, attractivity and the measure of non-compactness, as well as the measure of weak noncompactness. The tools of fractional calculus are found to be of great utility in improving the mathematical modeling of many natural phenomena and processes occurring in the areas of engineering, social, natural, and biomedical sciences. All abstract results in the book are illustrated by examples in applied mathematics, engineering, biomedical, and other applied sciences.
1. Preliminary Background
2. Caputo Fractional Difference Equations in Banach Spaces
3. Caputo Fractional Difference Inclusions
4. Ulam Stability for Fractional Difference Equations
5. Impulsive Fractional Difference Equations
6. Coupled Fractional Difference Systems
7. Coupled Caputo-Hadamard Fractional Differential Systems in Generalized Banach Spaces
8. Coupled Hilfer-Hadamard Fractional Differential Systems in Generalized Banach Spaces
9. Oscillation and Nonoscillation Results for Fractional q-Difference Equations and Inclusions
10. A Filippov's Theorem and Topological Structure of Solution Sets for Fractional q-Difference Inclusions
11. On Õ -Caputo Fractional Differential Equations in Banach Spaces
12. Ulam Stability for Õ -Caputo Fractional Differential Equations and Systems
13. Monotone Iterative Technique for psi-Caputo Fractional Differential Equations